Advective-diffusion equation

Nguen, N. and Reynen, J., 1984. A space-time least-squares finite element scheme for advection-diffusion equations. Cornput. Methods Appl Mech. Eng. 42, 331- 342. 

Biofilms adhere to surfaces, hence in nearly all systems of interest, whether a medical device or geological media, transport of mass from bulk fluid to the biofilm-fluid interface is impacted by the velocity field [24, 25]. Coupling of the velocity field to mass transport is a fundamental aspect of mass conservation [2]. The concentration of a species c(r,t) satisfies the advection diffusion equation... 

Combining the effects of molecular diffusion and pore-water advection yields a onedimensional advection-diffusion equation for a conservative solute ... 

If the solute imdergoes any chemical changes, a reaction term must be added to Eq. 12.4. In the absence of specific rate law information, diagenetic reactions are generally assumed to be first-order with respect to the solute concentration. Thus, the one-dimensional advection-diffusion equation far a nonconservative solute is given by... 

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

If particle mixing is assumed to be analogous to diffusion with sediment accumulation and radionuclide decay, the steady-state profile for excess activity of a nonexchangeable radionuclide is defined by an advective-diffusion equation. 

The pore scale boundary value problem is described by the advection-diffusion equation... 

Now it will become evident that for most oceanic circumstances k D, usually by several orders of magnitude, so that the molecular diffusi-vity can be safely ignored, giving the familiar time-dependent advection-diffusion equation... 

It is evident from inspection of Equation (25) that the mixing of water parcels with differing tritium concentrations will yield a disproportionate weighting toward the water with higher tritium. This can be quantified by considering the coupled tritium and He advection-diffusion equations ... 

Combining the above Equations (25)-(27), and (31) with some algebraic manipulation (Jenkins, 1987, 1998), one can derive an advection-diffusion equation for the tritium- He age as... 

In typical transport problems, one considers a scalar field 0, representing, for example, the concentration of some substance that evolves according to the advection-diffusion equation... 

For the well-mixed reservoirs, a conservation equation is written in which gain of 14C by inflow to the box (atmosphere or surface ocean) is balanced by the outflow to other boxes plus radioactive decay (see Section 2.8) of the tracer during its time in the reservoir. For the deep ocean, conservation is described by a partial differential advection-diffusion equation. The... 

The three-dimensional advection-diffusion equations for tracers, biological variables, have the form... 

Taking into account both advective and diffusive transport, the evolution of the concentration of a passive non-reactive substance in an incompressible flow is governed by the advection-diffusion equation... 

There is a connection between the Lagrangian representation based on advected particles and the Eulerian representation using concentration fields. As in the case of pure advection the solution of the advection-diffusion equation can be given in terms of trajectories of fluid elements. Equation (2.6) can be generalized for the diffusive case using the Feynman-Kac formula (see e.g. Durrett (1996)) as... 

L/U) is much smaller than the diffusion time over a cell, that is L2/D. As a result of the dominant advective transport the concentration across a single cell is almost uniform and the concentration difference between two adjacent cells is restricted to a narrow boundary layer in the vicinity of the separatrix. In this region the dominant balance in the advection-diffusion equation is between advection along the separatrix and diffusion across the separatrix through the boundary layer... 

Snapshots of a concentration field obtained numerically by solving the advection-diffusion equation with the velocity field (2.66) are shown in Fig. 2.16. 

In the context of combustion in non-premixed flames, this is called the Burke-Schumann solution. If the initial condition is such that T(x, t = 0) is not homogeneous in space, solving its advection-diffusion equation to obtain T(x, t) is needed in addition to solving the one for (x, t) before using (5.8)-(5.9) to obtain C(x, t). 

An additional time-dependent advection-diffusion equation is used to represent the mass transport of the cupric ions. This equation can be expressed as,... 

---